Question

Consider the line y=4 x -1 and the point P=(2,0). (a) Write the formula for a function d(x) that describes the distance between the point P and a point (x,y) on the line. You final answer should only involve the variable x. Then d(x) = √(4−x)2(4x−1)2 (b) d'(x)= (c) The critical number is x= . (d) The closest point on the line to P is ( , ).

Answer 1

a) d(x)=
`\sqrt{17x^(2) -12x+5}`

b)d'(x)=
`\frac{17x-6}{\sqrt{17x^(2) -12x+5} }`

c)The critical point is x=
`(6)/(17)`

d)Closest point is (
`(6)/(17)`,
`(7)/(17)`

Explanation:

We are given the line

`y=4x-1`

Let a point Q(
`x,y`) lie on the line.

Point P is given as P(2,0)

By distance formula, we have the distance D between any two points

A(
`x_(1),y_(1)`) and B(
`x_(2),y_(2)`) as

D=
`\sqrt{(x_(1)-x_(2))^2 + (y_(1)-y_2)^2}`

Thus,

d(x)=
`√((x-2)^2+(y-0)^2)`

But we have,
`y=4x-1`

So,

d(x)=
`√((x-2)^2+(4x-1)^2)`

Expanding,

d(x)=
`√(17x^2-12x+5)` - - - (a)

Now,

d'(x)=
`((d)/(dx) (17x^2-12x+5))/(2(√(17x^2-12x+5)) )`

i.e.

d'(x)=
`\frac{17x-6}{\sqrt{17x^(2) -12x+5} }` - - - (b)

Now, the critical point is where d'(x)=0

⇒
`\frac{17x-6}{\sqrt{17x^(2) -12x+5} }` =0

⇒
`x=(6)/(17)` - - - (c)

Now,

The closest point on the given line to point P is the one for which d(x) is minimum i.e. d'(x)=0

⇒
`x=(6)/(17)`

as
`y=4x-1`

⇒y=
`(7)/(17)`

So, closest point is (
`(6)/(17),(7)/(17)`) - - -(d)